Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre’s construction and then analyze the complexity. Ressayre’s construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering ≺ on R. The construction is clearly constructible over these objects. Each step looks effective, but there may be many steps. We produce an example of an exponential field R with a residue field section k and a well ordering ≺ on R such that is low and k and ≺ are Δ⁰₃, and Ressayre’s construction cannot be completed in .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-2-6,
author = {Paola D'Aquino and Julia F. Knight and Salma Kuhlmann and Karen Lange},
title = {Real closed exponential fields},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {163-190},
zbl = {1285.03036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-2-6}
}
Paola D'Aquino; Julia F. Knight; Salma Kuhlmann; Karen Lange. Real closed exponential fields. Fundamenta Mathematicae, Tome 219 (2012) pp. 163-190. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-2-6/